ScalingStacks

Definition 7 A graded $A$-module $M$ is a $\mathbb{Z}$-graded abelian group $M={\mathop{\oplus}\limits_{i\in\mathbb{Z}}}M_{i},$ together with group homomorphisms

	$\displaystyle X:$	$\displaystyle M_{i}\longrightarrow M_{i-2},$	$\displaystyle\hskip 14.45377pti\in\mathbb{Z},$		(183)
	$\displaystyle c:$	$\displaystyle M_{i}\longrightarrow M_{i+2},$	$\displaystyle\hskip 14.45377pti\in\mathbb{Z},$		(184)

that satisfy relations

$$Xc=cX\hskip 21.68121pt\mbox{and}\hskip 21.68121ptX^{2}=0.$$

(185)